The Plotkin powerdomain [dt-0067]
As before, let X, Y \in \mathcal {P}_f^\ast (\mathsf {K}(A)) be finite, non-empty sets of compact elements of a Scott domain A. The Plotkin powerdomain construction is based on the following preorder, simply combining the orderings from the Hoare and Smyth constructions: X \preceq _P Y\quad \text {iff}\quad X \preceq _H Y \land X \preceq _S Y This ordering is also called the Egli-Milner ordering. On a flat domain, it simplifies to: A\sqsubseteq B\quad \text {iff}\quad A = B \lor (\bot \in A \land (A \setminus \{\bot \})\subseteq B) Note that this is antisymmetric in the case of a flat domain, and the lub of a chain of sets can be found by taking the union of all elements of the chain if all elements contain \bot If not, then the element without \bot will be the least upper bound.
The failures of antisymmetry can only be observed in domains of height higher than one. Consider a set \{x,y\} containing two elements. According to the induced equivalence from this preorder \approx _P, this set would be considered equal to the set that contains those two elements plus any elements that lie between them on the information ordering: \{x,y\} \approx _P \{ z \mid x \sqsubseteq z \sqsubseteq y \} The Plotkin powerdomain is sometimes called the convex powerdomain, due to the similarity of this to the geometric definition of convexity. Thus, we use the ideal completion trick (as with the Hoare and Smyth constructions) here to arrive at a definition of powerdomain that distinguishes between all three programs outlined in Remark [dt-0063].